10.5 Group Decision Making

Often groups of people have to make decisions about what the group
will do. Societies are the classic example, where voting is used to
ascertain what the group wants. It may seem that voting is a good way
to determine what a group wants, and when there is a clear most-preferred choice, it is. However, there are major problems
with voting when there is not a clear preferred choice, as shown in
the following example.

Example 10.19:
Consider a purchasing agent that has to decide on a holiday destination for a
group of people, based on their preference. Suppose there are three
people, Alice, Bob and Cory, and three destinations, X, Y, and
Z. Suppose the agents have the following preferences, where >
means strictly prefers:

Alice: X>Y>Z.

Bob: Y>Z>X.

Cory: Z>X>Y.

Given these preferences, in a pairwise vote, X>Y because two out of the
three prefer X to Y. Similarly, in the voting, Y>Z and
Z>X. Thus, the preferences obtained by voting are not
transitive. This example is known as the Condorcet
paradox. Indeed, it is not clear what a group outcome should be in
this case, because it is symmetric between the outcomes.

A social preference function gives a preference relation for
a group. We would like a social preference function to depend on the
preferences of the individuals in the group.
It may seem that the Condorcet paradox is a problem with pairwise
voting; however, the following result shows that such paradoxes occur
with any social preference function.

Proposition.(Arrow's impossibility theorem)
If there are three or more outcomes, the following properties cannot
simultaneously hold for any social preference function:

Every individual preference that is complete and transitive is
allowed.

If every individual prefers outcome o1 to o2, the
group prefers o1 to o2.

The group preference between outcomes o1 and o2 depends
only on the individual preferences on o1 and o2 and not on the
individuals' preferences on other outcomes.

No individual gets to unilaterally decide the outcome (nondictatorship).

When building an agent that takes the individual preferences and gives a social
preference, we have to be aware that we cannot have all of these
intuitive and desirable properties. Rather than giving a group
preference that has undesirable properties, it may be better to point
out to the individuals how their preferences cannot be reconciled.